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Properties

Label 2-1100-220.43-c0-0-2
Degree $2$
Conductor $1100$
Sign $0.850 - 0.525i$
Analytic cond. $0.548971$
Root an. cond. $0.740926$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−1 + i)7-s + 8-s + i·9-s + i·11-s + (1 − i)13-s + (−1 + i)14-s + 16-s + (−1 − i)17-s + i·18-s + i·22-s + (1 − i)26-s + (−1 + i)28-s − 2i·31-s + 32-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−1 + i)7-s + 8-s + i·9-s + i·11-s + (1 − i)13-s + (−1 + i)14-s + 16-s + (−1 − i)17-s + i·18-s + i·22-s + (1 − i)26-s + (−1 + i)28-s − 2i·31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.548971\)
Root analytic conductor: \(0.740926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.792084601\)
\(L(\frac12)\) \(\approx\) \(1.792084601\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
11 \( 1 - iT \)
good3 \( 1 - iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23287057170661415242152260499, −9.439953413002062106782396390332, −8.361058663320756634252534582901, −7.45553995509924948313641411909, −6.58819782379565632812637479537, −5.75463057441446794277426385104, −5.05825656279048397881606670847, −4.01777393808750477548572985211, −2.83423699136721553216427975904, −2.14242648618276862370666185611, 1.42024358583790389868284908156, 3.22214521894513693028976208949, 3.68463061453536275729070645522, 4.53897738988138594217811797769, 6.00745923964873841552594307123, 6.49571802834505605315698371086, 6.99010028921841719034155289834, 8.365312532815018294243398181890, 9.130731628337644397572911566446, 10.25690638541201915646947168736

Graph of the $Z$-function along the critical line